= Solution
Fix $X_0=x$ and a finite horizon $T$. Begin under <Wiener measure> $\mathbb P$ with $X_t=x+W_t$, and let $K$ bound $|b|$. The bounded <progressively measurable process> $b(X)$ is an admissible <Itô integral> integrand. Put
$$
Z_T=\exp\left\{\int_0^T b(X_s)\,dW_s-\frac12\int_0^T b(X_s)^2ds\right\}.
$$
The <Novikov condition> follows from $\int_0^T b(X_s)^2ds\leq K^2T$. With $d\mathbb Q=Z_Td\mathbb P$, the <Girsanov theorem> says that $B=W-\int_0^\cdot b(X_s)ds$ is <Brownian motion>. Therefore \b[a <weak stochastic solution> exists], since
$$
\boxed{X_t=x+B_t+\int_0^t b(X_s)ds.}
$$
Consistent finite-horizon densities on canonical continuous path space give a solution for all finite times. No uniform integrability at infinite time is needed for this extension.
For \b[<uniqueness in law>], take any <weak stochastic solution> under a measure $\mathbb Q$ and reverse the <change of measure>:
$$
R_T=\exp\left\{-\int_0^T b(X_s)\,dB_s-\frac12\int_0^T b(X_s)^2ds\right\},\qquad d\mathbb P=R_Td\mathbb Q.
$$
Again the <Novikov condition> holds. Under $\mathbb P$, $W=B+\int b(X_s)ds=X-x$ is <Brownian motion>, so the law of $X$ is the fixed shifted <Wiener measure>. Substitution of $dB=dX-b(X)dt$ gives
$$
\boxed{R_T^{-1}=\exp\left\{\int_0^T b(X_s)\,dX_s-\frac12\int_0^T b(X_s)^2ds\right\}.}
$$
Under shifted <Wiener measure>, the <stochastic integral> on the right has a fixed <measurable> version as a functional of the canonical path. It can be defined by <simple predictable processes> converging in the integrand $L^2$ norm and an almost surely convergent subsequence of their integrals. Equivalence of the two finite-horizon measures preserves that version. Thus for every bounded <measurable> path functional $F$,
$$
\mathbb E_{\mathbb Q}F(X)=\mathbb E_{\text{shifted Wiener measure}}\left[F(X)\exp\left\{\int_0^T b(X_s)dX_s-\tfrac12\int_0^T b(X_s)^2ds\right\}\right].
$$
The right-hand side depends only on $b,x,T$, not on the chosen solution. This proves <uniqueness in law> on every finite horizon, and hence on the whole continuous path space. This is <Weak existence and uniqueness in law for an additive-noise SDE with bounded drift>. For merely <measurable> $b$, one should not assume that naive deterministic left-endpoint samples of $b(X)$ converge; the predictable $L^2$ approximation is the appropriate definition.
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